Rats problem

Long story short: I live in a terraced house and both my neighbours have a neglected garden with overgrown weeds (I mean it's way above the fence level) and waste all over the place. This has caused a rat infestation, and while they don't seem to be bothered (my neighbours) I am obviously affected by this: I contacted the council and after being bounced around I was told to contact the Derelict Site
I've provided all the necessary information and proof however I was told that (quoting) "there is no provision in the Derelict Sites Act , 1990 to deal with back gardens"

Now if I read this correctly:

The Derelict Sites Act defines a derelict site as any land that “detracts, or is likely to detract, to a material degree from the amenity, character or appearance of land in the neighbourhood of the land in question because of

• Structures which are in a ruinous, derelict or dangerous condition, or
• The neglected, unsightly or objectionable condition of the land or of structures on it, or
• The presence, deposit or collection of litter, rubbish, debris or waste

Those properties are not falling apart however they definitely fall in the category of:

• The neglected, unsightly or objectionable condition of the land or of structures on it, or
• The presence, deposit or collection of litter, rubbish, debris or waste

Is there anything I can do to force my neighbours to clean up their dump site?
Please don't suggest the use of a cat (I have a dog) or rat poison (done that but the source of the problem reside in my neighbours garden), I will contact HSE pest control but is there anything else that can be done from a legal point of view? Any other suggestion?